Real surfaces in complex surfaces : a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan
Slapar, Marko
Let ▫$S \subset X$▫ be a real compact surface, ▫${\mathcal C}^\infty$▫ embedded in a complex surface ▫$X$▫. The problem of existence of a regular Stein neighborhood basis on ▫$S$▫ in ▫$X$▫ has so far ...not been well understood. In a generic position, there were only finitely many complex points on ▫$X$▫, which can be classified as either elliptic or hyperbolic. By a result of Bishop, the nonexistence of elliptic complex points on ▫$S$▫ is a necessary condition for the existence of a Strin neighborhood basis of ▫$S$▫. We show that an embedded surface ▫$S$▫, without elliptic complex points, and with the extra condition of flatness at hyperbolic complex points, has a regular Stein neighborhood basis in ▫$X$▫. A connection between complex points and unions of totally real planes is then explored to prove a similar result for certain polynomially convex unions of totally real planes. Using these results, together with the global theory of complex points on embedded real surfaces, we give some new examples of totally real surfaces and Stein domains inside complex elliptic surfaces.
Type of material - dissertation
; adult, serious
Publication and manufacture - Ann Arbor : [M. Slapar], 2003
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